Download Sequences A sequence is a pattern of numbers - Kelvin-2011

Sequences
A sequence is a pattern of numbers; they
are useful for predicting future values
19 comes next because this
pattern increases by 4
3, 7, 11, 15, ... 19
18 comes next because this
pattern also increases by 4
2, 6, 10, 14, ... 18
Are they the SAME pattern? NO. Although they have the same
rule, they have different starting points
Each sequence consists of two things:
--a rule to find the next term
--a starting point
term number
1
2
3
4
5
3, 7, 11, 15, ... 19
term value
Term number refers to a term position
within the sequence (usually starting at
1)
Term value refers to the actual number
in the sequence
1
Example: Determine the patterns of the
following sequences:
(Be sure to specify both the starting
point and the rule)
1, 4, 7, 10, ...
Starts at: 1
Rule: increases by 3
3, 6, 12, 24, 48, ...
2, 5, 11, 23, ...
Starts at: 3
Rule: twice the previous value
Starts at: 2
Rule: twice the previous value plus 1
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Sequences can have many different kinds
of rules (as we've seen in those 3
examples)
BUT there are two main types:
arithmetic and geometric
Arithmetic Sequence
Follows a rule that adds/subtracts the
same amount each time
That amount is called the common
difference, or just d
If you know you have an arithmetic
sequence, how would you find the value of
d?
5, 11, 17, 23, ...
11 - 5 = 6 OR
23 - 17 = 6
What is d?
To find d, we can just take any term
value and subtract the previous term
value
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Geometric Sequences
Follows a rule where the same number is
multiplied to each term value to get the
next value
This number is called the common ratio,
or r
1, 3, 9, 27, ...
r=3
2, -4, 8, -16, 32, ...
r = -2
20, 10, 5, 5/2, 5/4, ...
r = 1/2
5 / 10 = 1/2 OR
5/2 / 5 = 1/2
To find r, we can just take any term value
and divide the previous term value
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So far, we've described sequences rules
in terms of what happens to the previous
value
(increases by 3)
1, 4, 7, 10, ...
What will the value of the 8th term be?
1
2
3
5
4
1, 4, 7, 10, ...
6
7
8
22
What will the value of 99th term be?
There is a formula that is based on
the term number (if you tell me
the term number, it gives back the
term value at that number)
tn = 3 * n - 2
t8 = 3 * 8 - 2 = 24 - 2 = 22
t99 = 3 * 99 - 2 = 297 - 2 = 295
This is an algebraic formula (we won't use these because
they are hard to come up with)... they are useful thought
because we can use them to find ANY term value
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When I asked to describe the
patterns... everyone described in a way
that built off of the previous value
(went up by 3, or doubled, etc...)
This is a natural way of thinking
However, it was not useful when we
wanted to predict the 99th term value
(it would take too long... we would have
to find the 98th, 97th, so on...)
If only there was a way to get the
calculator to compute the numbers for
us...
We could tell it "my pattern starts at 1,
and then to find the next value, just add
3 to the previous value"
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To work with sequences on the TI-83,
change the mode to SEQ
[MODE][FUNC --> SEQ]
To enter a new sequence 1, 4, 7, 10, ...
[Y=]
nMin = 1
u(n) = 3*n - 2
u(nMin) = 1
algebraic formula for this
sequence
value at term number 1
n = term number (nMin is just the
starting term number, usually 1)
u(nMin) = the term value at the first
term number
u is the name of the sequence
u(n) returns the term value at term
number n
To view the sequence that you've entered
[2nd][TABLE]
To view a specific term value at a given
term number n, from the home screen
u(n)
u(99) [ENTER] 295
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